A PROBABILISTIC REPRESENTATION FOR THE VALUE OF ZERO-SUM DIFFERENTIAL GAMES WITH INCOMPLETE INFORMATION ON BOTH SIDES

摘要

We prove that for a class of zero-sum differential games with incomplete information on both sides, the value admits a probabilistic representation as the value of a zero-sum stochastic differential game with complete information, where both players control a continuous martingale. A similar representation as a control problem over discontinuous martingales was known for games with incomplete information on one side (see [P. Cardaliaguet and C. Rainer, Math. Oper. Res., 34 (2009), pp. 769-794]), and our result is a continuous-time analogue of the so-called splitting game introduced in [R. Laraki, Internat. J. Game Theory, 30 (2002), pp. 359-376] and [S. Sorin, A First Course on Zero-Sum Repeated Games, Springer, New York, 2002] in order to analyze discrete-time models. It was proved by Cardaliaguet [SIAM J. Control Optim., 46 (2006), pp. 816-838; J. Math. Anal. Appl., 360 (2009), pp. 95-107] that the value of the games we consider is the unique solution of some Hamilton-Jacobi equation with convexity constraints. Our result provides therefore a new probabilistic representation for solutions of Hamilton Jacobi equations with convexity constraints as values of stochastic differential games with unbounded control spaces and unbounded volatility.